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We present distributed distance-based control (DDC), a novel approach for controlling a multi-agent system, such that it achieves a desired formation, in a resource-constrained setting. Our controller is fully distributed and only requires local state-estimation and scalar measurements of inter-agent distances. It does not require an external localization system or inter-agent exchange of state information. Our approach uses spatial- predictive control (SPC), to optimize a cost function given strictly in terms of inter-agent distances and the distance to the target location. In DDC, each agent continuously learns and updates a very abstract model of the actual system, in the form of a dictionary of three independent key-value pairs (~s, d), where d is the partial derivative of the distance measurements along a spatial direction ~s. This is sufficient for an agent to choose the best next action. We validate our approach by using DDC to control a collection of Crazyflie drones to achieve formation flight and reach a target while maintaining flock formation.